Problem 8 - College Algebra

Question

Determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 6 x-7 y+z &=2 \\ -x-y+3 z &=4 \quad \text { and }(4,2,-6) \\ 2 x+y-z &=1 \end{aligned} $$

Step-by-Step Solution

Answer

The ordered triple (4, 2, -6) is not a solution to the system.

Step 1: Substitute into the first equation

Take the ordered triple (4, 2, -6) and substitute the values into the first equation: \(6x - 7y + z = 2\). Here, \(x = 4\), \(y = 2\), and \(z = -6\).Substitute and simplify:\[6(4) - 7(2) + (-6) = 2\]Calculate the left side:\[24 - 14 - 6 = 4\]The left side equals 4, which is not equal to 2. Hence, the ordered triple does not satisfy the first equation.

Step 2: Verify solution conclusion

Since the ordered triple (4, 2, -6) does not satisfy the first equation, it is not necessary to check the remaining equations, because it already proves the triple is not a solution to the system.

Key Concepts

Ordered Triple

An "ordered triple" is a group of three specific numbers that are used to represent a solution to a system of three equations involving three variables. In

• the ordered triple
  • (4, 2, -6)
  • the first number corresponds to the variable \(x\)
  • the second number corresponds to \(y\)
  • the third number to \(z\)

This is similar to an ordered pair used in two-variable systems, but here a third value is included to account for the third variable. It is important that the order of numbers matches the variables in solution checking.

Equation Substitution

Equation substitution involves replacing the variables in the equation with specific values. These values come from the ordered triple in question. This is a critical step in verifying if this triple is indeed a solution to the system.For example:

• In the first equation, \(6x - 7y + z = 2\)
• plug in \(x = 4\), \(y = 2\), and \(z = -6\)
• simplify to check if the values satisfy the equation

If they do not satisfy even one of the equations, as seen in the solution above, the ordered triple is not a solution to the system. Substitution helps in clearly determining this without ambiguity.

Solution Verification

Solution verification is the process of ensuring whether the substituted values satisfy all the equations in the system. It involves calculating the left sides of the equations and comparing with the right sides it's supposed to equal. To perform solution verification:

• Equate and simplify both sides of the equation
• If results match, move to the next equation
• If any equation isn’t satisfied, as was the case here when left side was 4 but needed to be 2
• then the ordered triple is not a solution

It is a quick yet powerful step to determine if all equations in the system remain consistent with given values.

Algebraic Expressions

Algebraic expressions are key components of systems of equations and consist of numbers, variables, and mathematical operations.In each equation:

• the expressions \(6x - 7y + z\), \(-x - y + 3z\), and \(2x + y - z\) represent such algebraic expressions
• They include terms and coefficients, like \(6\) in \(6x\), which multiply the variables
• Expressions are simplified by performing calculations and combining like terms

Understanding algebraic expressions is crucial for effectively substituting, verifying, and solving systems of equations accurately.